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## G = (C4×S3)⋊F5order 480 = 25·3·5

### 3rd semidirect product of C4×S3 and F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — (C4×S3)⋊F5
 Chief series C1 — C5 — C15 — C3×D5 — C6×D5 — C6×F5 — D6⋊F5 — (C4×S3)⋊F5
 Lower central C15 — C30 — (C4×S3)⋊F5
 Upper central C1 — C4

Generators and relations for (C4×S3)⋊F5
G = < a,b,c,d,e | a4=b5=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b3, bd=db, be=eb, cd=dc, ece=a2c, ede=d-1 >

Subgroups: 820 in 152 conjugacy classes, 50 normal (36 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, C23, D5, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, Dic5, C20, C20, F5, D10, D10, C2×C10, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C42⋊C2, C4×D5, C4×D5, C2×Dic5, C2×C20, C2×F5, C2×F5, C22×D5, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, C3×F5, C3⋊F5, S3×D5, C6×D5, S3×C10, D30, C4×F5, C4×F5, C4⋊F5, C22⋊F5, C2×C4×D5, C422S3, D5×Dic3, S3×Dic5, D30.C2, D5×C12, S3×C20, C4×D15, C6×F5, C2×C3⋊F5, C2×S3×D5, D10.C23, D6⋊F5, Dic3⋊F5, C12×F5, C4×C3⋊F5, C4×S3×D5, (C4×S3)⋊F5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, F5, C4×S3, C22×S3, C42⋊C2, C2×F5, S3×C2×C4, C4○D12, C22×F5, C422S3, S3×F5, D10.C23, C2×S3×F5, (C4×S3)⋊F5

Smallest permutation representation of (C4×S3)⋊F5
On 120 points
Generators in S120
(1 49 19 34)(2 50 20 35)(3 46 16 31)(4 47 17 32)(5 48 18 33)(6 51 21 36)(7 52 22 37)(8 53 23 38)(9 54 24 39)(10 55 25 40)(11 56 26 41)(12 57 27 42)(13 58 28 43)(14 59 29 44)(15 60 30 45)(61 106 76 91)(62 107 77 92)(63 108 78 93)(64 109 79 94)(65 110 80 95)(66 111 81 96)(67 112 82 97)(68 113 83 98)(69 114 84 99)(70 115 85 100)(71 116 86 101)(72 117 87 102)(73 118 88 103)(74 119 89 104)(75 120 90 105)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)(81 82 83 84 85)(86 87 88 89 90)(91 92 93 94 95)(96 97 98 99 100)(101 102 103 104 105)(106 107 108 109 110)(111 112 113 114 115)(116 117 118 119 120)
(1 19)(2 16 5 17)(3 18 4 20)(6 23 7 25)(8 22 10 21)(9 24)(11 28 12 30)(13 27 15 26)(14 29)(31 48 32 50)(33 47 35 46)(34 49)(36 53 37 55)(38 52 40 51)(39 54)(41 58 42 60)(43 57 45 56)(44 59)(61 63 62 65)(66 68 67 70)(71 73 72 75)(76 78 77 80)(81 83 82 85)(86 88 87 90)(91 93 92 95)(96 98 97 100)(101 103 102 105)(106 108 107 110)(111 113 112 115)(116 118 117 120)
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)(61 71 66)(62 72 67)(63 73 68)(64 74 69)(65 75 70)(76 86 81)(77 87 82)(78 88 83)(79 89 84)(80 90 85)(91 101 96)(92 102 97)(93 103 98)(94 104 99)(95 105 100)(106 116 111)(107 117 112)(108 118 113)(109 119 114)(110 120 115)
(1 64)(2 65)(3 61)(4 62)(5 63)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)

G:=sub<Sym(120)| (1,49,19,34)(2,50,20,35)(3,46,16,31)(4,47,17,32)(5,48,18,33)(6,51,21,36)(7,52,22,37)(8,53,23,38)(9,54,24,39)(10,55,25,40)(11,56,26,41)(12,57,27,42)(13,58,28,43)(14,59,29,44)(15,60,30,45)(61,106,76,91)(62,107,77,92)(63,108,78,93)(64,109,79,94)(65,110,80,95)(66,111,81,96)(67,112,82,97)(68,113,83,98)(69,114,84,99)(70,115,85,100)(71,116,86,101)(72,117,87,102)(73,118,88,103)(74,119,89,104)(75,120,90,105), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120), (1,19)(2,16,5,17)(3,18,4,20)(6,23,7,25)(8,22,10,21)(9,24)(11,28,12,30)(13,27,15,26)(14,29)(31,48,32,50)(33,47,35,46)(34,49)(36,53,37,55)(38,52,40,51)(39,54)(41,58,42,60)(43,57,45,56)(44,59)(61,63,62,65)(66,68,67,70)(71,73,72,75)(76,78,77,80)(81,83,82,85)(86,88,87,90)(91,93,92,95)(96,98,97,100)(101,103,102,105)(106,108,107,110)(111,113,112,115)(116,118,117,120), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30)(31,36,41)(32,37,42)(33,38,43)(34,39,44)(35,40,45)(46,51,56)(47,52,57)(48,53,58)(49,54,59)(50,55,60)(61,71,66)(62,72,67)(63,73,68)(64,74,69)(65,75,70)(76,86,81)(77,87,82)(78,88,83)(79,89,84)(80,90,85)(91,101,96)(92,102,97)(93,103,98)(94,104,99)(95,105,100)(106,116,111)(107,117,112)(108,118,113)(109,119,114)(110,120,115), (1,64)(2,65)(3,61)(4,62)(5,63)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)>;

G:=Group( (1,49,19,34)(2,50,20,35)(3,46,16,31)(4,47,17,32)(5,48,18,33)(6,51,21,36)(7,52,22,37)(8,53,23,38)(9,54,24,39)(10,55,25,40)(11,56,26,41)(12,57,27,42)(13,58,28,43)(14,59,29,44)(15,60,30,45)(61,106,76,91)(62,107,77,92)(63,108,78,93)(64,109,79,94)(65,110,80,95)(66,111,81,96)(67,112,82,97)(68,113,83,98)(69,114,84,99)(70,115,85,100)(71,116,86,101)(72,117,87,102)(73,118,88,103)(74,119,89,104)(75,120,90,105), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120), (1,19)(2,16,5,17)(3,18,4,20)(6,23,7,25)(8,22,10,21)(9,24)(11,28,12,30)(13,27,15,26)(14,29)(31,48,32,50)(33,47,35,46)(34,49)(36,53,37,55)(38,52,40,51)(39,54)(41,58,42,60)(43,57,45,56)(44,59)(61,63,62,65)(66,68,67,70)(71,73,72,75)(76,78,77,80)(81,83,82,85)(86,88,87,90)(91,93,92,95)(96,98,97,100)(101,103,102,105)(106,108,107,110)(111,113,112,115)(116,118,117,120), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30)(31,36,41)(32,37,42)(33,38,43)(34,39,44)(35,40,45)(46,51,56)(47,52,57)(48,53,58)(49,54,59)(50,55,60)(61,71,66)(62,72,67)(63,73,68)(64,74,69)(65,75,70)(76,86,81)(77,87,82)(78,88,83)(79,89,84)(80,90,85)(91,101,96)(92,102,97)(93,103,98)(94,104,99)(95,105,100)(106,116,111)(107,117,112)(108,118,113)(109,119,114)(110,120,115), (1,64)(2,65)(3,61)(4,62)(5,63)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120) );

G=PermutationGroup([[(1,49,19,34),(2,50,20,35),(3,46,16,31),(4,47,17,32),(5,48,18,33),(6,51,21,36),(7,52,22,37),(8,53,23,38),(9,54,24,39),(10,55,25,40),(11,56,26,41),(12,57,27,42),(13,58,28,43),(14,59,29,44),(15,60,30,45),(61,106,76,91),(62,107,77,92),(63,108,78,93),(64,109,79,94),(65,110,80,95),(66,111,81,96),(67,112,82,97),(68,113,83,98),(69,114,84,99),(70,115,85,100),(71,116,86,101),(72,117,87,102),(73,118,88,103),(74,119,89,104),(75,120,90,105)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45),(46,47,48,49,50),(51,52,53,54,55),(56,57,58,59,60),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80),(81,82,83,84,85),(86,87,88,89,90),(91,92,93,94,95),(96,97,98,99,100),(101,102,103,104,105),(106,107,108,109,110),(111,112,113,114,115),(116,117,118,119,120)], [(1,19),(2,16,5,17),(3,18,4,20),(6,23,7,25),(8,22,10,21),(9,24),(11,28,12,30),(13,27,15,26),(14,29),(31,48,32,50),(33,47,35,46),(34,49),(36,53,37,55),(38,52,40,51),(39,54),(41,58,42,60),(43,57,45,56),(44,59),(61,63,62,65),(66,68,67,70),(71,73,72,75),(76,78,77,80),(81,83,82,85),(86,88,87,90),(91,93,92,95),(96,98,97,100),(101,103,102,105),(106,108,107,110),(111,113,112,115),(116,118,117,120)], [(1,9,14),(2,10,15),(3,6,11),(4,7,12),(5,8,13),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30),(31,36,41),(32,37,42),(33,38,43),(34,39,44),(35,40,45),(46,51,56),(47,52,57),(48,53,58),(49,54,59),(50,55,60),(61,71,66),(62,72,67),(63,73,68),(64,74,69),(65,75,70),(76,86,81),(77,87,82),(78,88,83),(79,89,84),(80,90,85),(91,101,96),(92,102,97),(93,103,98),(94,104,99),(95,105,100),(106,116,111),(107,117,112),(108,118,113),(109,119,114),(110,120,115)], [(1,64),(2,65),(3,61),(4,62),(5,63),(6,66),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(13,73),(14,74),(15,75),(16,76),(17,77),(18,78),(19,79),(20,80),(21,81),(22,82),(23,83),(24,84),(25,85),(26,86),(27,87),(28,88),(29,89),(30,90),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,97),(38,98),(39,99),(40,100),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120)]])

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J ··· 4N 5 6A 6B 6C 10A 10B 10C 12A 12B 12C ··· 12L 15 20A 20B 20C 20D 30 60A 60B order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 ··· 4 5 6 6 6 10 10 10 12 12 12 ··· 12 15 20 20 20 20 30 60 60 size 1 1 5 5 6 30 2 1 1 5 5 6 10 10 10 10 30 ··· 30 4 2 10 10 4 12 12 2 2 10 ··· 10 8 4 4 12 12 8 8 8

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 8 8 8 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 S3 D6 D6 C4○D4 C4×S3 C4×S3 C4○D12 F5 C2×F5 C2×F5 C2×F5 D10.C23 S3×F5 C2×S3×F5 (C4×S3)⋊F5 kernel (C4×S3)⋊F5 D6⋊F5 Dic3⋊F5 C12×F5 C4×C3⋊F5 C4×S3×D5 S3×Dic5 D30.C2 S3×C20 C4×D15 C4×F5 C4×D5 C2×F5 C3×D5 Dic5 C20 D5 C4×S3 Dic3 C12 D6 C3 C4 C2 C1 # reps 1 2 2 1 1 1 2 2 2 2 1 1 2 4 2 2 8 1 1 1 1 4 1 1 2

Matrix representation of (C4×S3)⋊F5 in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 50 0 0 0 0 0 0 50 0 0 0 0 0 0 50 0 0 0 0 0 0 50
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 60 60 60 60
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 60 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 60 0 0
,
 47 0 0 0 0 0 59 13 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 9 31 0 0 0 0 23 52 0 0 0 0 0 0 54 0 47 47 0 0 14 7 14 0 0 0 0 14 7 14 0 0 47 47 0 54

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,50],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,1,0,0,60,0,0,0,1,0,60,0,0,0,0,1,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,60,0,0,0,0,0,0,0,60,0],[47,59,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,23,0,0,0,0,31,52,0,0,0,0,0,0,54,14,0,47,0,0,0,7,14,47,0,0,47,14,7,0,0,0,47,0,14,54] >;

(C4×S3)⋊F5 in GAP, Magma, Sage, TeX

(C_4\times S_3)\rtimes F_5
% in TeX

G:=Group("(C4xS3):F5");
// GroupNames label

G:=SmallGroup(480,985);
// by ID

G=gap.SmallGroup(480,985);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,100,1356,9414,2379]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^5=c^4=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^3,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^2*c,e*d*e=d^-1>;
// generators/relations

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